https://neckpinch.com/authors/vestislav-apostolov/New arXiv papers by Vestislav Apostolov, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2509.01639arxiv:2509.01639Mon, 01 Sep 2025 00:00:00 GMTVestislav Apostolov, Giuseppe Barbaro, Jeffrey Streets, Yury Ustinovskiy — We establish a local equivalence between toric steady Kähler-Ricci solitons and A-type toric generalized Kähler-Ricci solitons (GKRS). Under natural global conditions we show this equivalence extends to complete GKRS, yielding a general construction of new examples in all dimensions. We show that in four dimensions, all GKRS are either described by the generalized Kähler Gibbons-Hawking ansatz, or have split tangent bundle, or are A-type toric. This yields a local classification in four dimensions, together with a conjecturally exhaustive construction of complete symplectic-type examples.Ricci solitonsKähler–Ricci solitonsKähler geometry https://arxiv.org/abs/2412.02564arxiv:2412.02564Tue, 03 Dec 2024 00:00:00 GMTVestislav Apostolov, Abdellah Lahdili, Eveline Legendre — We show that if X is a smooth Fano manifold which caries a Kähler Ricci soliton, then the canonical cone of the product of X with a complex projective space of sufficiently large dimension is a Calabi–Yau cone. This can be seen as an asymptotic version of a conjecture by Mabuchi and Nikagawa. This result is obtained by the openness of the set of weight functions v over the momentum polytope of a given smooth Fano manifold, for which a v-soliton exists. We discuss other ramifications of this approach, including a Licherowicz type obstruction to the existence of a Kähler Ricci soliton and a Fujita type volume bound for the existence of a v-soliton.Ricci solitonsKähler–Ricci solitonsRicci-flat & Calabi–YauKähler geometryKähler–Einstein & K-stability https://arxiv.org/abs/2404.14595arxiv:2404.14595Mon, 22 Apr 2024 00:00:00 GMTVestislav Apostolov, Jeffrey Streets, Yury Ustinovskiy — Building on works of Boulanger and Goto, we show that Goto's scalar curvature is the moment map for an action of generalized Hamiltonian automorphisms of the associated Courant algebroid, constrained by the choice of an adapted volume form. We derive an explicit formula for Goto's scalar curvature, and show that it is constant for generalized Kähler-Ricci solitons. Restricting to the generically symplectic type case, we realize the generalized Kähler class as the complexified orbit of the Hamiltonian action above. This leads to a natural extension of Mabuchi's metric and K-energy, implying a conditional uniqueness result. Finally, in this setting we derive a Calabi-Matsushima-Lichnerowicz obstruction and a Futaki invariant.Ricci solitonsKähler–Ricci solitonsKähler geometryScalar curvatureStability & uniquenessKähler–Einstein & K-stability https://arxiv.org/abs/2305.15626arxiv:2305.15626Thu, 25 May 2023 00:00:00 GMTVestislav Apostolov, Charles Cifarelli — For each partition of the positive integer n= ℓ +∑ⱼ₌₁^ℓ dⱼ, where ℓ≥ 1 and dⱼ ≥ 0 are integers, we construct a continuous (ℓ−1)-parameter family of explicit complete gradient steady Kähler–Ricci solitons on ℂⁿ admitting a hamiltonian 2-form of order ℓ and symmetry group rm U(d₁+ 1) × ⋯ × rm U(d_ℓ + 1). For ℓ=1, d₁=n-1 we obtain Cao's example [17] whereas for other partitions the metrics are new. Furthermore, when n=2, ℓ=2, d₁=d₂=0 we obtain complete gradient steady Kähler–Ricci solitons on ℂ² which have positive sectional curvature but are not isometric to Cao's rm U(2)-invariant example. This disproves a conjecture by Cao. We also present a construction yielding explicit families of complete gradient steady Kähler-Ricci solitons on ℂⁿ containing higher dimensional extensions of the Taub-NUT Ricci-flat Kähler metric on ℂ². When n≥ 3, the complete Ricci-flat Kähler metrics, and when n≥ 2, their deformations to complete gradient steady Kähler Ricci solitons seem not to have been observed before our work.Ricci solitonsKähler–Ricci solitonsRicci-flat & Calabi–YauSphere theoremsKähler geometry https://arxiv.org/abs/2211.09104arxiv:2211.09104Wed, 16 Nov 2022 00:00:00 GMTVestislav Apostolov, Xin Fu, Jeffrey Streets, Yury Ustinovskiy — We formulate an extension of the Calabi conjecture to the setting of generalized Kähler geometry. We show a transgression formula for the Bismut Ricci curvature in this setting, which requires a new local Goto/Kodaira-Spencer deformation result, and use it to show that solutions of the generalized Calabi-Yau equation on compact manifolds are classically Kähler, Calabi-Yau, and furthermore unique in their generalized Kähler class. We show that the generalized Kähler-Ricci flow is naturally adapted to this conjecture, and exhibit a number of a priori estimates and monotonicity formulas which suggest global existence and convergence. For initial data in the generalized Kähler class of a Kähler Calabi-Yau structure we prove the flow exists globally and converges to this unique fixed point. This has applications to understanding the space of generalized Kähler structures, and as a special case yields the topological structure of natural classes of Hamiltonian symplectomorphisms on hyperKähler manifolds. In the case of commuting-type generalized Kähler structures we establish global existence and convergence with arbitrary initial data to a Kähler, Calabi-Yau metric, which yields a new d d^c-lemma for these structures.Ricci flowKähler–Ricci flowGeneralized & G₂ flowsRicci-flat & Calabi–YauKähler geometryLong-time existence & convergence https://arxiv.org/abs/2109.10295arxiv:2109.10295Tue, 21 Sep 2021 00:00:00 GMTVestislav Apostolov, Jeffrey Streets, Yury Ustinovskiy — Under broad hypotheses we derive a scalar reduction of the generalized Kähler-Ricci soliton system. We realize solutions as critical points of a functional analogous to the classical Aubin energy defined on the orbit of a natural Hamiltonian action of diffeomorphisms, thought of as a generalized Kähler class. This functional is convex on a large set of paths in this space, and using this we show rigidity of solitons in their generalized Kähler class. As an application we prove uniqueness of the generalized Kähler-Ricci solitons on Hopf surfaces constructed in arXiv:1907.03819, finishing the classification in complex dimension 2.Ricci solitonsKähler–Ricci solitonsKähler geometryStability & uniqueness